ESAIM: Probability and Statistics

Table of Contents - November 2005 - Volume 9  

Research Article

Risk bounds for mixture density estimation

Rakhlin, Alexandera1, Panchenko, Dmitrya2 and Mukherjee, Sayana3

a1 Center for Biological and Computational Learning, Massachusetts Institute of Technology, Cambridge, MA 02139, USA; rakhlin@mit.edu

a2 Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02143, USA.

a3 Institute of Statistics and Decision Sciences, Institute for Genome Sciences and Policy, Duke University, Durham, NC 27708, USA.

Abstract

In this paper we focus on the problem of estimating a bounded density using a finite combination of densities from a given class. We consider the Maximum Likelihood Estimator (MLE) and the greedy procedure described by Li and Barron (1999) under the additional assumption of boundedness of densities. We prove an $O(\frac{1}{\sqrt{n}})$ bound on the estimation error which does not depend on the number of densities in the estimated combination. Under the boundedness assumption, this improves the bound of Li and Barron by removing the $\log n$ factor and also generalizes it to the base classes with converging Dudley integral.

(Received July 21 2004)

(Online publication November 15 2005)

Key Words:

  • Mixture density estimation;
  • maximum likelihood;
  • Rademacher processes.

Mathematics Subject Classification:

  • 62G05;
  • 62G07;
  • 62G20
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